Optimal. Leaf size=22 \[ \frac {\sqrt {a x^2} \sqrt {1+x^2}}{x} \]
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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {15, 267}
\begin {gather*} \frac {\sqrt {x^2+1} \sqrt {a x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 267
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2}}{\sqrt {1+x^2}} \, dx &=\frac {\sqrt {a x^2} \int \frac {x}{\sqrt {1+x^2}} \, dx}{x}\\ &=\frac {\sqrt {a x^2} \sqrt {1+x^2}}{x}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a x^2} \sqrt {1+x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 19, normalized size = 0.86
method | result | size |
gosper | \(\frac {\sqrt {a \,x^{2}}\, \sqrt {x^{2}+1}}{x}\) | \(19\) |
default | \(\frac {\sqrt {a \,x^{2}}\, \sqrt {x^{2}+1}}{x}\) | \(19\) |
risch | \(\frac {\sqrt {a \,x^{2}}\, \sqrt {x^{2}+1}}{x}\) | \(19\) |
meijerg | \(\frac {\sqrt {a \,x^{2}}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{2}+1}\right )}{2 x \sqrt {\pi }}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 19, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a} x^{2} + \sqrt {a}}{\sqrt {x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 18, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a x^{2}} \sqrt {x^{2} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 17, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a x^{2}} \sqrt {x^{2} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.33, size = 19, normalized size = 0.86 \begin {gather*} {\left (\sqrt {x^{2} + 1} \mathrm {sgn}\left (x\right ) - \mathrm {sgn}\left (x\right )\right )} \sqrt {a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.67, size = 19, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a}\,\sqrt {x^2+1}\,\sqrt {x^2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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